Ideal Rigid and Free Terminations

• At a fixed end, the string's transverse displacement is zero. Consider the general traveling wave solution to the wave equation: $y(t, x) = y^{+}(ct - x) + y^{-}(ct + x)$. If the string is fixed at $x = 0$, then $y(t, 0) = 0$ and $y^{+}(ct) = -y^{-}(ct)$, which indicates that displacement traveling waves reflect from a fixed end with an inversion (or a reflection coefficient of -1).

• Since the displacement at a rigid termination is always zero, the physical velocity must also be zero for all time. Therefore, traveling-wave components of velocity will also reflect with a reflection coefficient of -1 at such a boundary.

• Force and velocity traveling-wave components are related by the wave impedance as $f^{+} = Rv^{+}, f^{-} = -Rv^{-}$. At a rigid terminiation, $v^{+} = -v^{-}$ (from above). Thus, force wave components can be related at a rigid termination as:
  $$f^{+} = R v^{+} = R (-v^{-}) = -R v^{-} = f^{-}.$$
  From this, we see that traveling-wave components of force are related by a reflection coefficient of +1 at a rigid termination.

• At a free end, $\partial y/\partial x = 0$ because no transverse force is possible. At such a boundary, traveling-wave components of force must reflect with a coefficient of -1. To determine the reflection coefficient for displacement waves, we first note that force waves are proportional to the string slope. Differentiation of our general traveling-wave solution by $x$ leads to (see this link):
  $$\frac{\partial}{\partial x} y(t,x) = 0 = -\frac{1}{c} \frac{\part... ...tial t}y^{+}(t - x/c) + \frac{1}{c} \frac{\partial}{\partial t}y^{-}(t + x/c),$$
  After integrating this expression with respect to time, we have $y^{+}(t) = y^{-}(t)$ at $x = 0$, indicating that displacement traveling waves reflect with a coefficient of +1.